L(2) SOLVABILITY AND REPRESENTATION BY CALORIC LAYER POTENTIALS IN TIME-VARYING DOMAINS

We consider boundary value problems for the heat equation in time-vary ing graph domains of the form Omega = {(x(0), x, t) is an element of R x R(n-1) x R: x(0) > A(x, t)}, obtaining solvability of the Dirichlet and Neumann problems when the data lie in L(2)(partial derivative Ome ga). We also prove optimal regularity estimates for solutions to the D irichlet problem when the data lie in a parabolic Sobolev space of fun ctions having a tangential (spatial) gradient, and one half of a time derivative in L(2)(partial derivative Omega). Furthermore, we obtain r epresentations of our solutions as caloric layer potentials. We prove these results for functions A(x, t) satisfying a minimal regularity co ndition which is essentially sharp from the point of view of the relat ed singular integral theory. We construct counterexamples which show t hat our results are in the nature of ''best possible.''